Quadrilateral

A quadrilateral is a closed shape that is formed by joining four points among which any three points are non-collinear. In simple words, a quadrilateral is a polygon with 4 sides, 4 angles, and 4 vertices. Let us learn more about the quadrilateral shape, the properties of quadrilaterals, the different types of quadrilaterals along with a few quadrilateral examples.

A quadrilateral is a polygon with four sides, four angles and four vertices. Whenever we name a quadrilateral, we need to keep in mind the order of the vertices. For example, the following quadrilateral should be named ABCD, BCDA, ADCB, or, DCBA. It cannot be named as ACBD or DBAC, since they change the order of vertices in which a quadrilateral is formed. The following quadrilateral ABCD has four sides: AB, BC, CD, DA, and two diagonals: AC and BD.

Meaning of Quadrilateral

The word 'Quadrilateral' is derived from a Latin word, in which, 'Quadra' means four and 'Latus' means sides. It should be noted that all 4 sides of a quadrilateral may or may not be equal. There are different types of quadrilaterals and they are uniquely identified on the basis of their distinct properties.

Although a quadrilateral always has four sides, four angles, and four vertices, the measure of the sides and angles differ. It is to be noted that the sum of the interior angles of a quadrilateral is always equal to 360°. The following table lists the different types of quadrilaterals.

Each of the quadrilaterals discussed above has its own properties. Though, there are some properties that are common to all quadrilaterals. They are as follows.

• All quadrilaterals have four sides.
• All quadrilaterals have four vertices.
• All quadrilaterals have two diagonals.
• The sum of the interior angles of quadrilaterals is 360°.

Now, let us read about the other properties of different quadrilaterals in detail. We can identify a quadrilateral by using the following properties of quadrilaterals.

Square

A square is a quadrilateral with four equal sides and four right angles.

Observe the square given above and relate it to the following properties:

• A square has 4 equal sides. Here, AB = BC = CD = DA
• It 4 right angles. Here, ∠ A= ∠B = ∠C = ∠D = 90°
• It has 2 pairs of parallel sides. Here, AB ∥ DC and AD ∥ BC
• It has 2 equal diagonals. Here, AC = BD
• It has diagonals that are perpendicular to each other. Here, AC ⊥ BD and the diagonals bisect each other.

Rectangle

A rectangle is a quadrilateral in which the opposite sides are equal and parallel and each of its interior angles is 90°.

Observe the rectangle given above and relate it to the following properties:

• A rectangle has 2 pairs of parallel sides. Here, AB ∥ DC and AD ∥ BC
• It has 4 right angles. Here, ∠A = ∠B = ∠C = ∠D = 90°
• The opposite sides of a rectangle are equal. Here, AB = DC and AD = BC
• It has 2 equal diagonals. Here, AC = BD and the diagonals bisect each other.

Parallelogram

A parallelogram is a quadrilateral in which the opposite sides are parallel.

Observe the parallelogram given above and relate it to the following properties:

• A parallelogram has 2 pairs of parallel sides. Here, PQ ∥ RT and PR ∥ QT
• The opposite sides of a parallelogram are equal. Here, PQ = RT and PR = QT
• The opposite angles of a parallelogram are equal. Here, ∠P = ∠T and ∠Q = ∠R
• It has 2 diagonals that bisect each other.

Trapezium

A trapezium is a quadrilateral in which one pair of opposite sides is parallel.

Observe the trapezium given above and relate it to the following properties:

• In a trapezium, the sides that are parallel to each other are called bases. Here, EF and GH are the bases.
• The sides that are not parallel to each other are called legs. Here, EG and FH are the legs.

There is nothing special about the sides, angles, or diagonals of a trapezium. But if the two non-parallel opposite sides are of equal length, then it is called an isosceles trapezium. The following quadrilateral XYZW is an isosceles trapezium, in which the legs are equal, i.e., WX = ZY, and the diagonals are also equal, i.e., XZ = WY.

Rhombus

A rhombus is a quadrilateral with four equal sides.

Observe the rhombus given above and relate it to the following properties:

• A rhombus has 2 pairs of parallel sides. Here, EH ∥ FG and EF ∥ HG
• It has 4 equal sides. Here, EH = HG = GF = FE
• The opposite angles of a rhombus are equal. Here, ∠E = ∠G and ∠H = ∠F
• It has diagonals that are perpendicular to each other. Here, EG ⊥ HF and the diagonals bisect each other.

Kite

A kite is a quadrilateral in which two pairs of adjacent sides are equal.

Observe the properties of a kite given below and relate it to the figure given above.

• A kite has 2 pairs of equal adjacent sides. Here, AB = BC and CD = DA
• It has one pair of opposite angles (which are obtuse) that are equal. Here, ∠A = ∠C
• It has diagonals that are perpendicular to each other. Here, AC ⊥ BD
• The longer diagonal bisects the shorter diagonal.

Think Tank

• Can a kite be called a parallelogram?
• What elements of a trapezium should be changed to make it a parallelogram?

Area of Quadrilaterals

The area of a quadrilateral is the number of unit squares that can be fit into it. The following table lists the formulas for finding the area of quadrilaterals.

☛Topics Related to Quadrilaterals

Check out some interesting articles related to quadrilaterals.

• Geometric Shapes
• Triangles
• Euclidean Geometry
• Solid Shapes
• Angles
• Parallelograms
• Angle Sum Property in Quadrilaterals

What is a Quadrilateral in Maths?

A quadrilateral is a closed two-dimensional figure that has 4 sides, 4 angles, and 4 vertices. A few examples of quadrilaterals are square, rectangle, kite, and trapezium.

What are the Different Types of Quadrilaterals?

There are different types of quadrilaterals that are identified on the basis of their unique properties. For example, square, rectangle, parallelogram, rhombus, kite, trapezium, isosceles trapezium are all categorized under quadrilaterals.

What is the Sum of the Interior Angles in a Quadrilateral?

In any type of quadrilateral, the sum of the interior angles is always equal to 360°. For example, a rectangle is a quadrilateral with each of its interior angles equal to 90° which makes it (90 × 4) = 360°.

What are the Common Properties of all Quadrilaterals?

Though there are different types of quadrilaterals, they share a few properties that are common. They are listed as follows:

• All quadrilaterals have four sides.
• They have four vertices.
• They have two diagonals.
• The sum of all interior angles is 360°.

How to Find the Area of a Quadrilateral?

The area of a quadrilateral is the space occupied by it. Since each quadrilateral has its own unique properties, their area is calculated using different formulas. However, it is to be noted that the area of a quadrilateral is always expressed in square units. A few examples of quadrilaterals are square and rectangle. The area of a square of side 'a' is calculated by the formula: Area = 'a × a' or a² and the area of a rectangle whose length is 'l' and width is 'w' is calculated by the formula: Area = 'l × w'.

How to Find the Perimeter of a Quadrilateral?

The perimeter of a quadrilateral is the total length of its boundary. As we know that a quadrilateral has 4 sides, the perimeter of a quadrilateral can be found by adding all the sides of the quadrilateral. For example, if a rectangle has a length of 6 units and a width of 4 units then we use the formula for the perimeter of a rectangle which is: 2(length + width). Substituting the values in the formula, we get 2 (6 + 4) = 20 units.

What is the Sum of the Angles of a Quadrilateral?

The sum of the interior angles of a quadrilateral is always 360°. This rule applies to all quadrilaterals like the square, rectangle, trapezium, kite, rhombus, and so on.

Is a Quadrilateral a Parallelogram?

Yes, a quadrilateral can be a parallelogram if its opposite sides are parallel. However, a quadrilateral is not always necessarily a parallelogram, it can also be a trapezium or a kite. This is because a quadrilateral is defined as any polygon that has four sides, four angles and four vertices.

Are all Sides of a Quadrilateral Equal?

All sides of a quadrilateral may not always be necessarily equal. In some cases, if all sides of a quadrilateral are equal, then that particular quadrilateral is identified as a square or a rhombus.